Chapter 9, Problem 38SE

LetRbe a quasi-ordering and let S be the relation on the set of equivalence classes of $R\cap R^{-1}$such that (C,D) belongs to 5, whereCandDare equivalence classes ofR, if and only if there are elementscofCanddofDsuch that (c, d) belongs toR. Show thatSis a partial ordering.

LetLbe a lattice. Define the meet $(\wedge )$and join $(\vee )$operations by $x\wedge y=\mathrm{glb}\left(x,y\right)$and $x\vee y=\mathrm{lub}\left(x,y\right)$.